David S. Dummit and Richard M. Foote

Some courses in mathematics have a single textbook that has become an almost canonical choice — Rudin's analysis book, Munkres' Topology book, and Ahlfors' complex analysis are extremely common choices as one flips through syllabi. There does not seem to be any such canonical choice for an abstract algebra textbook, though there are several contenders for that title. One of these books, Abstract Algebra by David Dummit and Richard Foote, has just released a third edition.

The topics covered in this book are, if not completely standard, also certainly nothing surprising. The first six chapters deal with groups: basic definitions, examples, the Sylow theorems, group actions, and a variety of related topics. The next three chapters define rings, and discuss the basic properties of rings and domains. One of these chapters is devoted exclusively to polynomial rings — and in this chapter the authors introduce Grobner bases, the topic which really distinguishes this edition from previous ones. The next three chapters deal with modules and vector spaces before moving on to a chapter on field theory and a separate complete chapter on Galois theory.

The last four chapters consist of topics which I would consider beyond the scope of most one year courses in algebra — commutative rings, discrete valuation rings, Dedekind domains, and a bit of algebraic geometry, in addition to nice introductions to homological algebra, group cohomology, and the representation theory of finite groups. An appendix contains some category theory, enough to placate those users who want to learn the material, while not enough to distract from the main thrust of the book. In the introduction, the authors give several suggestions as to what material one might include in a one-year course, though one of the appeals of a book such as this one is that the professor can pick and choose from a wide variety of topics.

As mentioned above, the main addition to this edition of the book is the introduction of several sections on Grobner bases. The authors revisit the topic several times after initially introducing the concepts, and unlike many later editions of books, these additions flow quite well with the rest of the material. In particular, the exposition and the exercises are of as high a quality as one would expect from a textbook that is as ubiquitous as this one. Certainly there are other Algebra books that might be more appropriate for some audiences — this book is far from the easiest book on the market, and also far from the most technical — and there are topics that I think are presented better in Gallian or Hungerford — but there is no doubt that this book is one of the best on the market in what it does, and anyone looking for an algebra textbook should give Dummit and Foote a serious look.